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Supports of certain infinitely divisible probability measures on locally convex spaces. (English) Zbl 0776.60008

Summary: Let \(B\) be a separable Banach space and let \(\mu\) be a centered Poisson probability measure on \(B\) with Lévy measure \(M\). Assume that \(M\) admits a polar decomposition in terms of a finite measure \(\sigma\) on the unit sphere of \(B\) and a Lévy measure \(\rho\) on \((0,\infty)\). The main result of this paper provides a complete description of the structure of \({\mathcal S}_ \mu\), the support of \(\mu\). Specifically, it is shown that: (i) if \(\int_{(0,1]}s \rho(ds)=\infty\), then \({\mathcal S}_ \mu\) is a linear space and is equal to the closure of the semigroup generated by \({\mathcal S}_ M\) (the support of \(M)\) and the negative of the barycenter of \(\sigma\); and (ii) if \(\int_{(0,1]}s \rho(ds)<\infty\) and zero is in the support of \(\rho\), then \({\mathcal S}_ \mu\) is a convex cone and is equal to the closure of the semigroup generated by \({\mathcal S}_ M\). The result (i) yields an affirmative answer to the question, open for some time, of whether the support of a stable probability measure of index \(1 \leq \alpha<2\) on \(B\) is a translate of a linear space. Analogs of these results, for both Poisson and stable probability measures defined on general locally convex spaces, are also provided.

MSC:

60B11 Probability theory on linear topological spaces
60E07 Infinitely divisible distributions; stable distributions
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